Guess what? IT WORKS.
How a cold-blooded animal can get a fever is beyond me, but that's beside the point.
Anyway, this setup uses a forgotten beast with toxic secretions, meaning it should produce the contaminant constantly. The pump on the left moves water through diagonal fortifications to avoid the pressure-teleport effect that prevents flow. the water comes from an aquifer channel hidden by the axle. It picks up the toxin and flows out through a grate directly to the right of the FB, since flow can sometimes push creatures through fortifications but not grates. Currently a bug exists where a building destroyer may not attack a building directly next to it, which can be taken advantage of here if you're not fortunate enough to get an artifact grate. I'm not certain but catching sight of a creature might interrupt this behavior, so be careful.
The toxin ends up covering the tile with a roller on a ramp (selected in second picture). The pump there is just to power the roller without leaking.
Here's a side view schematic of the cart track:
# = fortification W= stationary cart (the heavier, the better to prevent water backflow from pushing it)
L = exploit curve track ramp (the two directions of the track are against a wall and in the direction of acceleration)
_ LLLLLL W#
/ \_______
The ramp on the bottom left of the schematic is the one covered with poisoned water. The "hill" between this and the other trench does not have a track on the flat tile on top of it, to facilitate derailing.
The cart is pushed out of the water trench by the roller and will fly over the gap onto the curve track ramps where it picks up speed. It will miss the first few tiles of them due to being airborne. After it picks up enough speed to launch a glob, it strikes the stationary cart and falls into the gap. The rollers push it back up over the hill, where it becomes airborne, hits the rear wall, and falls back into the toxic water to pick up another load.
The water laced with forgotten beast extract [833] strikes a creature and falls through the floor, which consists of down stairs covered by bars. It doesn't appear to leave a coating of the substance but the effects are felt anyway. I'm not sure if having standing water in the room prevents infection or not. If you use this setup, use grates, not bars. Building destroyers immediately dismantle bars but have to wear grates down first.
This setup is only for the use of one cart, they will get jammed if you use more. Trying to use this with a flooded multiple-cart "magazine" could be tricky, since you need to get the toxin onto the tiles where the carts fill up.